Explore the complexities of real numbers as infinite decimals in this 51-minute lecture from the Insights into Mathematics series. Delve into three distinct approaches to defining real numbers, focusing on the popular concept of infinite decimals as sequences of arbitrarily chosen digits. Examine the advantages of this approach, including its ability to transform processes into numbers, while also confronting its limitations in specifying general numbers and defining arithmetic operations. Investigate the role of the Axiom of Choice in providing an axiomatic framework for real numbers as infinite choice decimals. Gain insights into historical debates surrounding these mathematical concepts and understand the challenges they pose to the foundations of arithmetic.